Structure computation and discrete logarithms in finite abelian p-groups

@article{Sutherland2011StructureCA,
  title={Structure computation and discrete logarithms in finite abelian p-groups},
  author={Andrew V. Sutherland},
  journal={Math. Comput.},
  year={2011},
  volume={80},
  pages={477-500}
}
We present a generic algorithm for computing discrete logarithms in a finite abelian p-group H, improving the Pohlig—Hellman algorithm and its generalization to noncyclic groups by Teske. We then give a direct method to compute a basis for H without using a relation matrix. The problem of computing a basis for some or all of the Sylow p-subgroups of an arbitrary finite abelian group G is addressed, yielding a Monte Carlo algorithm to compute the structure of G using O(|G| 1/2 ) group operations… 

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References

SHOWING 1-10 OF 42 REFERENCES

On some computational problems in finite abelian groups

TLDR
These algorithms are based on a modification of Shanks' baby-step giant-step strategy, and have the advantage that their computational complexity and storage requirements are relative to the actual order, discrete logarithm, or size of the group, rather than relative to an upper bound on the group order.

A space efficient algorithm for group structure computation

  • Edlyn Teske
  • Computer Science, Mathematics
    Math. Comput.
  • 1998
TLDR
A new algorithm is presented for computing the structure of a finite abelian group, which has to store only a fixed, small number of group elements, independent of the group order, and it is proved that the expected run time is O(√n) and the O-constants are determined.

Groups St Andrews 1997 in Bath, I: A polynomial-time theory of black box groups I

TLDR
It is demonstrated that the “nonabelian normal structure” of matrix groups over finite fields can be mapped out in great detail by polynomial-time randomized (Monte Carlo) algorithms.

Order computations in generic groups

TLDR
It is proved that a generic algorithm can compute |α| for all α ∈ S ⊆ G in near linear time plus the cost of a single order computation with N = λ(S), and it is shown that in most cases the structure of an abelian group G can be determined using an additional O (Nδ/4 ) group operations, given an O ( Nδ ) bound on |G| = N.

Computing the structure of a finite abelian group

TLDR
An algorithm is presented that computes the structure of a finite abelian group G from a generating system M and executes O(|M|√|G|) group operations and stores O(√ |G |) group elements.

Lower Bounds for Discrete Logarithms and Related Problems

  • V. Shoup
  • Computer Science, Mathematics
    EUROCRYPT
  • 1997
TLDR
Lower bounds on the complexity of the discrete logarithm and related problems are proved that match the known upper bounds: any generic algorithm must perform Ω(p1/2) group operations, where p is the largest prime dividing the order of the group.

The expected number of random elements to generate a finite abelian group

TLDR
It is shown that the expected number of elements from G (chosen independently and with the uniform distribution) so that the elements chosen generate G is less than r, the Riemann zeta-function constant.

Quadratic class numbers and character sums

TLDR
An explicit version of Burgess' theorem valid for prime discriminants is given and, as an application, the class number of a 32-digit discriminant is computed.

On taking roots in finite fields

TLDR
The main result is shown that finding the least x such that x2 = a MOD(m) is NP-complete (even if m is factored).

Fast Exponentiation with Precomputation (Extended Abstract)

TLDR
This paper presents a practical method of speeding up cryptographic systems using precomputed values to reduce the number of multiplications needed, and allows the computation of gn for n < N in O(log N/log log N) group multiplications.